On the behavior of meromorphic functions at the ideal boundary of a Riemann surface

Abstract:

In a former work the author established an analog of a classical theorem of Painlevé in the context of an arbitrary resolutive compactification of a Riemann surface. In the same setting, a refinement of the argument used in the above yields an elementary proof of a theorem of Riesz-Luzin-Privaloff type: If a meromorphic function $f$ tends to zero at each point of a subset $E$ of the ideal boundary and $E$ has positive harmonic measure, then $f \equiv 0$ on $R$. The well-known inclusion relations ${U_{HB}} \subset {\mathcal {O}_{M{B^{\ast }}}}$ and ${U_{HD}} \sim {\mathcal {O}_{M{D^{\ast }}}}$, are then established from the point of view of the resolutivity of the Wiener and Royden compactification respectively.